I don't have direct experience with United (or moving to the UK with pets directly) but when we brought our cats over to Stockholm in December, SAS (partner with United at least in Star Alliance) required the APHIS 7001 as a health certificate. I am looking at it right now and it says that it is the "United States Interstate and International Certificate of Health Examination for Small Animals" It looks like we had the vet fill it out though, with their microchip numbers, vaccination history, etc. And the vet signed it too with her accreditation information. This was in addition to the other, longer form that the USDA needs to put the raised stamp on (Annex 2), but the USDA didn't stamp our APHIS 7001.
Here is some other info online:
http://www.aphis.usda.gov/regulations/vs/iregs/animals/animal_faq.shtml#threeThe airline didn't actually take certificate from us, nor did the entry officers. Then again it all struck us as a bit lackadaisical - from my understanding it seems like the UK may be quite a bit more strict than Sweden.
I could swear I read another thread on here with someone with specific United experience from within the last year though. May want to have a check.